Any orientation can be achieved from an initial orientation aligned with the space frame by rotating about some unit axis by a particular angle. We call the unit axis omega-hat and the rotation distance theta. If we multiply these two together, we get the 3-vector omega-hat theta. This is a 3-parameter representation of orientation. We call these 3 parameters the exponential coordinates representing the orientation of one frame relative to another. This is an alternative representation to a rotation matrix.

We call these exponential coordinates because of the connection to linear differential equations. In particular, we should view omega-hat as an angular velocity that is followed for theta seconds, and we have to integrate the angular velocity from the initial orientation to find the final orientation.

Before solving that problem, let's look at a familiar problem in linear ordinary differential equations in a single variable: x-dot = a times x, where a is a constant. The solution, as you learn in any course on differential equations, is e to the a t times x at time zero, where the exponential function e to the a t is defined by the series expansion shown here.

This scalar linear differential equation has an analogous vector linear differential equation, where x is now an n-vector and A is a constant n by n matrix. The solution to this differential equation has the same form as the single-variable case. The term e to the A t is called a matrix exponential. As we'll see in the next video, this equation can be used to integrate an angular velocity, where the matrix A is the 3 by 3 skew-symmetric representation of the angular velocity.